Norman Foo wrote:
> Formally, I think Fritz is right about the necessity of higher order
> concepts in semantics. In a 1st-order logic, compactness forces on
> us the results that finiteness, standardness for numbers, etc. are
> not expressible. If one tries to circumvent these by inventing
> predicates for them, these new predicates will in their turn have
> non-standard interpretations.
Maybe I'm missing something, but the debate between the first-orderists
and the higher-orderists seems to me to be a nonstarter; the two
positions are symmetrical. The first-orderist is going to insist that
she just *means* (bang the table, stamp the foot) the set of natural
numbers by the predicate "natural number" together with the usual
first-order axiomatization, regardless of the fact that any such
axiomatization has nonstandard interpretations. The higher-orderist
argues that the issue is moot for him: the semantics for his
higher-order quantifiers *guarantees* that by "natural number" (defined
in some familiar higher-order manner) he means the natural numbers. But
here is where the situation seems directly analogous to the
first-orderist's: what guarantees that those higher-order quantifiers
are interpreted according to the intended higher-order semantics?
What's to prevent them from being interpreted by means of Henkin's
generalized "higher-order" models? No additional axioms can force the
one interpretation over the other. But then the higher-order theory can
just be interpreted as a first-order theory in higher-order guise, and
hence is just as open to nonstandard interpretations. The
higher-orderist can only reply that when he uses his higher-order
quantifiers he *means* (bang the table, stamp the foot) full-blown
higher-order quantification; he *intends* to be quantifying over the
full power set of his domain. But the higher-orderist's intentions here
are nailed down no better (and no worse) than the first-orderists.
Insofar as the higher-orderist is allowed to rest content that his
quantifiers have their intended meanings, the first-orderist should be
no less allowed to rest content that his natural number predicate has
*its* intended meaning. They are in the same semantic boat.
If this is right, higher-order semantics provides no practical
advantage over first-order for knowledge interchange. The sentences
involved in a knowledge interchange cannot carry their intended
semantics on their sleeve. Nonstandard interpretations are possible
either way, and no number of axioms of any order are going to rule
them out. But is this so bad? We certainly seem no worse off in the
context of knowledge interchange than in everyday communication.
Nothing guarantees that you don't have some nonstandard understanding
of the term `natural number' when I'm talking to you; we could always
systematically reinterpret one another's words to fit our different
semantics. So in knowledge interchange we do just as we do every day:
as Quine put it, we simply acquiesce in our background language, i.e.,
we take certain concepts as given and understood and go from there.
(I take it this was roughly Tom Gruber's point in a recent post.) If
we're first-orderists, we take, e.g., "natural number" to pick out the
natural numbers and, on that assumption, define such notions as
finitude in the manner that John Sowa pointed out; we just don't mean
to include nonstandard "numbers" in the interpretation of "natural
number". If we're higher-orderists we take our higher-order
quantifiers to range over the power set of our domain and, on that
assumption, define the natural numbers, finitude, etc.; we just don't
mean anything less than the full power set as the range of our
quantifiers. The best we can do in knowledge interchange, then, is to
make our intended semantics as explicit as possible, assume our
interlocutors understand us and proceed with the interchange. As in
everyday communication, that'll probably be plenty good enough.
A final point. The above argument seems to leave first-orderism and
higher-orderism on the same footing. But that seems misleading. In the
first-order case, completeness tells us that our proof theory--the stuff
we can program into a computer (modulo well known limitations)--and our
notion of logical entailment match up perfectly; anything entailed by
some set of assumptions is provable from that set and (by soundness)
vice versa. In the higher-order case logical truth and entailment so
vastly outstrip any possible proof theory that we don't even get close;
we will never, in principle, capture the connection between a set of
assumptions and its higher-order logical consequences proof
theoretically. Given their parity in other respects, surely this gives
us at least some reason to prefer a first-order over a higher-order
semantics for an interlingua.
--Chris Menzel
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Christopher Menzel | Internet --> cmenzel@tamu.edu
Associate Professor, Philosophy | Phone -----> (409) 845-8764
Texas A&M University | Fax -------> (409) 845-0458
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